Low-order-refined (LOR) preconditioners [1]}, [2]} are well
known to be spectrally equivalent [3]} for problems in \(H(\operatorname{grad})\) .
For problems in \(H(\operatorname{curl})\)  and \(H(\operatorname{div})\) , LOR preconditioners have been studied in
[4]}, [5]}. The auxiliary low-order problem is sparse, but
even for the \(H(\operatorname{grad})\)  case (REF ) devising efficient relaxations is
challenging [6]}. This is because the low-order-refined grid is
anisotropic, and pointwise smoothers on the auxiliary low-order problem become
ineffective as \(p\)  increases. To overcome this, Pazner [6]} applies
patchwise multigrid with ICC relaxations to (REF ), relying on a good
ordering of the DOFs. This method has not been studied for ()
or (), to the best of our knowledge. Instead, Pazner, Kolev &
Dohrmann apply the AMS and ADS algebraic multigrid solvers of Hypre
[8]}, [9]}, [10]} to the auxiliary low-order problem, which
implement the strategy proposed by Hiptmair & Xu [11]}. These LOR
approaches do not naturally combine with static condensation, since the density
of the Schur complement arising from the elimination of DOFs that are interior
to the original grid counteracts the sparsity offered by the low-order problem.
